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Full text of "Treatise On Analysis Vol-Ii"

170       XIII    INTEGRATION

converges. Note that if m2 g N < (m + I)2, we have

, .     m2
'<*)-„-/.ğ<>

and deduce that the sequence (/N(X))NğI tends to 0 for almost all x e [0,1] (with
respect to Lebesgue measure). Hence show that, for almost all x e [0,1], the sequence
(An*— [An;t]) is equirepartitioned with respect to Lebesgue measure (Section 13.4,
Problem 7).

16. Let U be a continuous endomorphism of the space LR(X, p) with norm <^ 1 (in other
words, such that NI(£/ •/) <£ NI(/)) and such that the relation/^ 0 implies U-f^O
(13.6). If/is any function in the class f, we denote by U • /any function in the class

u-f.

(a) Let (fn) be a sequence of functions in Ğ^R which converges almost everywhere
to a function/and is such that there exists a function h e 3?^ satisfying \fn\ ^ /z for
all n. Show that under these conditions the sequence (U-fn) converges almost every-
where to £/*/ (Reduce to the case /= 0, and consider the sequence of functions

(b)   For each finite sequence (tk)i;^k^n of real numbers, put

, ...,/„) >0, then

•S-n('l,...sO =    SUP   (*i + -
l<fc<n

Show that if sn(tlf . . . , tn) > 0, then for all /B+1 6 R we have
(1)                                 *i + sn(t2 , . . . , tn+1) ^ 5n(/i, . . . , tn).

(c)    For each function / e ^R , show that

sJiU-f, . . . , £/n+1 •/) < t/ - JH(/, ...,(/" •/)

almost everywhere. (Use (1) and the fact that for any n functions fl9 ...,/„ 6 &^
we have

sup   U • fk ^ U '   sup fk

l^k^n                      l^fc^n

almost everywhere.)

(d)    Let E(/) be the measurable set of points x e X such that

/(*) + (U-f)(x) + • - • + (t/"-1 •/)(*) > 0
for some integer n > 0. Show that

> 0

~-

f

JE(/)

(E. Hopf's maximal ergodic theorem). (For each integer n ^> 1, let En(/) denote the
set of x e X such that

/(*) + ((/•/)(*) + • • - + (£/* • /)W > o